A note on the structure of $\frac{\mathbb{F}_qS_5}{J(\mathbb{F}_qS_5)}$

Abstract

A complete algebraic structure of $\frac{\mathbb{F}_qS_5}{J(\mathbb{F}_qS_5)}$, where $\mathbb F_q=\mathbb F_{p^n}$ is a finite field with characteristic $p$ (a prime) and $J(\mathbb{F}_qS_5)$ is the Jacobson radical of the group algebra $\mathbb{F}_qS_5$ is given.

Keywords

• Group algebra
• Wedderburn decomposition
• unit group

References

1. A. A. Bovdi and A. Szakacs, The unitary subgroup of the multiplicative group of the modular group algebra of a finite abelian $p$-group, Math. Zametki, 45(6) (1989), 23-29.
2. L. Creedon and J. Gildea, Unitary units of the group algebra $\mathbb{F}_{2^k}Q_8$, Internat. J. Algebra Comput., 19(2) (2009), 283-286.
3. C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Pure and Applied Mathematics, $XI$, Interscience Publishers (a division of John Wiley \& Sons, Inc.), New York-London, 1962.
4. A. Dholakia, Introduction to Convolutional Codes with Applications, Kluwer Academic Publishers, Boston-London, 1994.
5. R. A. Ferraz, Simple components of center of $FG/J(FG)$, Comm. Algebra, 36(9) (2008), 3191-3199.
6. J. Gildea, The structure of the unit group of the group algebra $\mathbb{F}_{2^{k}}A_{4}$, Czechoslovak. Math. J., 61(2) (2011), 531-539.
7. J. Gildea and F. Monaghan, Units of some group algebras of groups of order 12 over any finite field of characteristic 3, Algebra Discrete Math., 11(1) (2011), 46-58.
8. T. Hurley, Group rings and ring of matrices, Int. J. Pure Appl. Math., 31(3) (2006), 319-335.

9. T. Hurley, Convolutional codes from units in matrix and group rings, Int. J. Pure Appl. Math., 50(3) (2009), 431-463.
10. G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, 682, Springer, Berlin, 1978.
11. G. Karpilovsky, The Jacobson Radical of Group Algebras, North-Holland Mathematics Studies, 135, North-Holland Publishing Co., Amsterdam, 1987.
12. M. Khan, R. K. Sharma and J. B. Srivastava, The unit group of $\mathbb{F}S_4$, Acta Math. Hungar., 118 (2008), 105-113.
13. Y. Kumar, R. K. Sharma and J. B. Srivastava, The structure of the unit group of the group algebra $\mathbb{F}S_5$ where $\mathbb{F}$ is a finite field with char$(\mathbb{F})=p>5$, Acta Math. Acad. Paedagog. Nyhazi, 33(2) (2017), 187-191.
14. S. Maheshwari and R. K. Sharma, The unit group of group algebra $\mathbb{F}_qSL(2,\mathbb Z_3)$, J. Algebra Comb. Discrete Struct. Appl., 3(1) (2016), 1-6.
15. N. Makhijani, R. K. Sharma and J. B. Srivastava, A note on units in $\mathbb{F}_{p^m}D_{2p^n}$, Acta Math. Acad. Paedagog. Nyhazi, 30(1) (2014), 17-25.
16. N. Makhijani, R. K. Sharma and J. B. Srivastava, The unit group of algebra of circulant matrices, Int. J. Group Theory, 3(4) (2014), 13-16.
17. N. Makhijani, R. K. Sharma and J. B. Srivastava, The unit group of finite group algebra of a generalized dihedral group, Asian-Eur. J. Math., 7(2) (2014), 1450034 (5 pp).
18. N. Makhijani, R. K. Sharma and J. B. Srivastava, Units in $\mathbb{F}_{2^{k}}D_{2n}$, Int. J. Group Theory, 3(3) (2014), 25-34.
19. N. Makhijani, R. K. Sharma and J. B. Srivastava, A note on the structure of $\mathbb{F}_{p^{k}}A_{5}/J(\mathbb{F}_{p^{k}}A_{5})$, Acta Sci. Math. (Szeged), 82 (2016), 29-43.
20. C. Polcino Milies and S. K. Sehgal, An Introduction to Group Rings, Algebra and Applications, 1, Kluwer Academic Publishers, Dordrecht, 2002.
21. R. K. Sharma, J. B. Srivastava and M. Khan, The unit group of $\mathbb{F}S_3$, Acta Math. Acad. Paedagog. Nyhazi, 23(2) (2007), 129-142.
22. R. K. Sharma, J. B. Srivastava and M. Khan, The unit group of $\mathbb{F}A_4$, Publ. Math. Debrecen, 71 (2007), 21-26.
23. R. K. Sharma and P. Yadav, The unit group of $\mathbb{Z}_p Q_8$, Algebras Groups Geom., 25(4) (2008), 425-429.
24. R. K. Sharma and P. Yadav, Unit group of algebra of circulant matrices, Int. J. Group Theory, 2(4) (2013), 1-6.